Investigating exact density-functional theory of a model semiconductor

Using the diffusion quantum Monte Carlo method, we calculte the ground-state density and energy, and the quasiparticle band gap, of a model semiconductor. The exchange-correlation potential of density-functional theory (DFT), Vxc(r), is obtained using optimization techniques. From this we calculate the DFT functionals Esc and Tₛ and the DFT band gap for various external potentials and compare the results with the local-density approximation (LDA). Whereas energies are found to be very accurate in the LDA, and the density reasonably good, we find large differences in the shape of Vxc(r).